Paper Review

Summary

1. Use gaussian processes (GPs) to disentangle contradictory reports of planets versus rotation artefacts from Kapteyn’s star.
2. Observed radial velocities best explained by star rotation only
3. Be wary of using sinusoids for modelling quasi-periodicity

Gaussian Process

• mean function, \(\mu(t)\)
• covariance or kernel function, \(k(t_i, t_j)\)

\[\boldsymbol{Y} = \begin{bmatrix} Y_1 \\ \vdots \\ Y_n \end{bmatrix} \sim \mathcal{GP}(\boldsymbol{\mu}, \boldsymbol{\Sigma})\]

where \(\boldsymbol{\mu} = \mu(t_i),\; \boldsymbol{\Sigma} = \mathrm{Cov}(Y_i, Y_j) = k(t_i, t_j)\quad i,j = 1, \dots, n\).

\[k(\tau; \lambda) = A \exp\left\{-\left( \frac{t_i - t_j}{\lambda}\right)^2 - \Gamma \sin^2 \left(\frac{\pi (t_i - t_j)}{P}\right) \right\} + \sigma_i^2 \delta_{ij}\]

where \(A\) is the amplitude, \(\lambda\) is the correlation timescale, \(\Gamma\) is a smoothness parameter, \(P\) is the rotation period, and \(\sigma\) is the white noise.

Dataset

• 92 HARPS spectra (Pepe et al. 2000)
• 20 HIRES spectra (Vogt et al. 1994)
• Across 10.1y

Methodology I

1. Fit a GP using different kernels (QPK, QPK + SE)

• Model 1 (quasi-periodic)
• Model 2 (quasi-periodic + long-term drift in H\(\alpha\))

2. Hyperparameters fitted using maximum likelihood estimation, priors from literature

• GPs implemented in george package in R
• MCMC sampling done using emcee package in R

Methodology II

3. Model 1 vs Model 2 compared using BIC (lower is better) and Likelihood Ratio statistic.

4. Fit third model of RVs using above GP + Keplerian orbit

5. Examine the residuals after fitting to see any presence of a signal.

Results

Conclusions

• Can use GPs to jointly model H\(\alpha\) and RV simultaneously.

• QP GP returns a rotational period of 124.7 days

• No residuals signals suggesting no exoplanet

Limitations

• Very sparse data!

• GP perhaps have overfit; can consider a simpler less flexible kernel.